Problem: Noah borrows $\$2000$ from his father and agrees to repay the loan and any interest determined by his father as soon as he has the money. The relationship between the amount of money, $A$, in dollars that Noah owes his father (including interest), and the elapsed time, $t$, in years, is modeled by the following equation. $A=2000e^{0.1t}$ How long did it take Noah to pay off his loan if the amount he paid to his father was equal to $\$2450$ ? Give an exact answer expressed as a natural logarithm.
Explanation: Thinking about the problem We want to know how many years, $t$, it took for Noah to repay his debt, $A$, of $\$2450$. So we need to find the value of $t$ for which $A=2450$. Substituting $2450$ in for $A$ in the model gives us the following equation. $2450=2000e^{0.1t}$ Solving the equation We can solve the equation as shown below. $\begin{aligned}2000e^{0.1t}&=2450\\\\ e^{0.1t}&=1.225\\\\ 0.1t&=\ln\left(1.225\right)\\\\ t&=10\cdot{{\,\ln\left(1.225\right)}}\\\\ \end{aligned}$ It will take $10\cdot {{\,\ln\left(1.225\right)}}$ years for Noah to repay his loan. The expression above represents an exact solution to the problem. We can use a calculator to approximate the value of the expression, but this will be a rounded inexact answer. The answer The answer is $10\cdot{{\ln\left(1.225\right)}}$ years.